Theorem fliftf 5740

Description: The domain and range of the function F. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	|- F = ran ( x e. X |-> <. A , B >. )
flift.2	|- ( ( ph /\ x e. X ) -> A e. R )
flift.3	|- ( ( ph /\ x e. X ) -> B e. S )

Assertion

Ref	Expression
fliftf	|- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )

Distinct variable groups:   x, R   ph, x   x, X   x, S

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem fliftf

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 109	. . . . 5 |- ( ( ph /\ Fun F ) -> Fun F )
2		flift.1	. . . . . . . . . . 11 |- F = ran ( x e. X |-> <. A , B >. )
3		flift.2	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> A e. R )
4		flift.3	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> B e. S )
5	2, 3, 4	fliftel 5734	. . . . . . . . . 10 |- ( ph -> ( y F z <-> E. x e. X ( y = A /\ z = B ) ) )
6	5	exbidv 1802	. . . . . . . . 9 |- ( ph -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
7	6	adantr 274	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
8		rexcom4 2732	. . . . . . . . 9 |- ( E. x e. X E. z ( y = A /\ z = B ) <-> E. z E. x e. X ( y = A /\ z = B ) )
9		19.42v 1883	. . . . . . . . . . . 12 |- ( E. z ( y = A /\ z = B ) <-> ( y = A /\ E. z z = B ) )
10		elisset 2723	. . . . . . . . . . . . . 14 |- ( B e. S -> E. z z = B )
11	4, 10	syl 14	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. X ) -> E. z z = B )
12	11	biantrud 302	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X ) -> ( y = A <-> ( y = A /\ E. z z = B ) ) )
13	9, 12	bitr4id 198	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( E. z ( y = A /\ z = B ) <-> y = A ) )
14	13	rexbidva 2451	. . . . . . . . . 10 |- ( ph -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
15	14	adantr 274	. . . . . . . . 9 |- ( ( ph /\ Fun F ) -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
16	8, 15	bitr3id 193	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z E. x e. X ( y = A /\ z = B ) <-> E. x e. X y = A ) )
17	7, 16	bitrd 187	. . . . . . 7 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. x e. X y = A ) )
18	17	abbidv 2272	. . . . . 6 |- ( ( ph /\ Fun F ) -> { y | E. z y F z } = { y | E. x e. X y = A } )
19		df-dm 4589	. . . . . 6 |- dom F = { y | E. z y F z }
20		eqid 2154	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
21	20	rnmpt 4827	. . . . . 6 |- ran ( x e. X |-> A ) = { y | E. x e. X y = A }
22	18, 19, 21	3eqtr4g 2212	. . . . 5 |- ( ( ph /\ Fun F ) -> dom F = ran ( x e. X |-> A ) )
23		df-fn 5166	. . . . 5 |- ( F Fn ran ( x e. X |-> A ) <-> ( Fun F /\ dom F = ran ( x e. X |-> A ) ) )
24	1, 22, 23	sylanbrc 414	. . . 4 |- ( ( ph /\ Fun F ) -> F Fn ran ( x e. X |-> A ) )
25	2, 3, 4	fliftrel 5733	. . . . . . 7 |- ( ph -> F C_ ( R X. S ) )
26	25	adantr 274	. . . . . 6 |- ( ( ph /\ Fun F ) -> F C_ ( R X. S ) )
27		rnss 4809	. . . . . 6 |- ( F C_ ( R X. S ) -> ran F C_ ran ( R X. S ) )
28	26, 27	syl 14	. . . . 5 |- ( ( ph /\ Fun F ) -> ran F C_ ran ( R X. S ) )
29		rnxpss 5010	. . . . 5 |- ran ( R X. S ) C_ S
30	28, 29	sstrdi 3136	. . . 4 |- ( ( ph /\ Fun F ) -> ran F C_ S )
31		df-f 5167	. . . 4 |- ( F : ran ( x e. X |-> A ) --> S <-> ( F Fn ran ( x e. X |-> A ) /\ ran F C_ S ) )
32	24, 30, 31	sylanbrc 414	. . 3 |- ( ( ph /\ Fun F ) -> F : ran ( x e. X |-> A ) --> S )
33	32	ex 114	. 2 |- ( ph -> ( Fun F -> F : ran ( x e. X |-> A ) --> S ) )
34		ffun 5315	. 2 |- ( F : ran ( x e. X |-> A ) --> S -> Fun F )
35	33, 34	impbid1 141	1 |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 2125   {cab 2140   E.wrex 2433   C_ wss 3098   <.cop 3559   class class class wbr 3961   |-> cmpt 4021   X. cxp 4577   dom cdm 4579   ran crn 4580   Fun wfun 5157   Fn wfn 5158   -->wf 5159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-fv 5171

This theorem is referenced by:  qliftf  6554